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CSE554Cell ComplexesSlide 7 Cells The boundary of a k-cell (k>0) has dimension k-1 – Examples: A 1-cell is bounded by two 0-D points A 2-cell is bounded by a 1-D curve A 3-cell is bounded by a 2-D surface 0-cell1-cell 2-cell3-cell (boundary is colored blue)

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CSE554Cell ComplexesSlide 12 Representing the object as a cell complex – Approach 2: create a 0-cell for each object pixel (voxel), and connect them to form higher dimensional cells. Reproducing 4-connectivity in 2D and 6-connectivity in 3D Cell Complex from Binary Pic

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CSE554Cell ComplexesSlide 14 Algorithm: Approach 1 Handling cells shared by adjacent pixels (voxels) – Create an array of integers that keeps track of whether a cell is created (e.g., at a pixel, at a pixel’s edge, or at a pixel’s point), and if it is created, the index of the cell. – Look up this array before creating a new cell 0 0000 0 000 0 000 0 0 000 0 0 0 0 0 00 0 00 00 0 00 0 0 Index of 2-cell Index of 1-cell Index of 0-cell

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CSE554Cell ComplexesSlide 16 Algorithm: Approach 2 2D: – Create a 0-cell at each object pixel, a 1-cell for two object pixels sharing a common edge, and a 2-cell for four object pixels sharing a common point 3D: – Create a 0-cell at each object voxel, a 1-cell for two object voxels sharing a common face, a 2-cell for four object voxels sharing a common edge, and a 3- cell for eight object voxels sharing a common point Follow the note in Approach 1 for storing cell indices

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CSE554Cell ComplexesSlide 19 Simple Pairs How can we remove cells from a complex without changing its topology?

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CSE554Cell ComplexesSlide 20 Simple Pairs Definition – A pair {x, y} such that y is on the boundary of x, and there is no other cell in the complex with y on its boundary. x y {x, y} is a simple pair y {x,y} is not a simple pair x

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CSE554Cell ComplexesSlide 21 Simple Pairs Definition – A pair {x, y} such that y is on the boundary of x, and there is no other cell in the complex with y on its boundary. x y {x, y} is a simple pair x y

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CSE554Cell ComplexesSlide 22 Simple Pairs Definition – A pair {x, y} such that y is on the boundary of x, and there is no other cell in the complex with y on its boundary. – In a simple pair, x is called a simple cell, and y is called the witness of x. A simple cell can pair up with different witnesses x y {x, y} is a simple pair y y x y

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CSE554Cell ComplexesSlide 24 Exhaustive Thinning Removing all simple pairs in parallel at each iteration – Only the topology of the cell complex is preserved – If a simple cell has multiple witnesses, an arbitrary choice is made // Exhaustive thinning on a cell complex C 1.Repeat: 1.Let S be all simple pairs in C 2.If S is empty, Break. 3.Remove all cells in S from C 2.Output C

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CSE554Cell ComplexesSlide 25 Exhaustive Thinning Removing all simple pairs in parallel at each iteration – Only the topology of the cell complex is preserved 2D example

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CSE554Cell ComplexesSlide 26 Exhaustive Thinning Removing all simple pairs in parallel at each iteration – Only the topology of the cell complex is preserved 3D example

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CSE554Cell ComplexesSlide 27 Exhaustive Thinning Removing all simple pairs in parallel at each iteration – Only the topology of the cell complex is preserved A more interesting 2D shape

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CSE554Cell ComplexesSlide 28 Exhaustive Thinning Removing all simple pairs in parallel at each iteration – Only the topology of the cell complex is preserved A 3D shape

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CSE554Cell ComplexesSlide 29 Isolated cells A cell x is isolated if it is not on the boundary of other cells – A k-dimensional skeleton is made up of isolated k-cells Isolated cells are colored

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CSE554Cell ComplexesSlide 33 Medial Cells (2D) For a 1-cell x that is isolated during thinning: – I(x), R(x) measures the “thickness” and “length” of shape – A greater difference means the local shape is more “elongated” I R R-I 1-I/R

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CSE554Cell ComplexesSlide 34 Medial Cells (3D) For a 2-cell x that is isolated during thinning: – I(x), R(x) measures the “thickness” and “width” of shape – A greater difference means the local shape is more “plate-like” I R R-I: face size 1-I/R: face color

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CSE554Cell ComplexesSlide 35 Medial Cells (3D) For a 1-cell x that is isolated during thinning: – I(x), R(x) measures the “width” and “length” of shape – A greater difference means the local shape is more “tubular” R-I: edge length 1-I/R: edge color I R

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CSE554Cell ComplexesSlide 36 Medial Cells and Thinning A cell x is a medial cell if it is isolated and the difference between R(x) and I(x) exceeds given thresholds – A pair of absolute/relative difference thresholds is needed for medial cells at each dimension 2D: thresholds for medial 1-cells – t1 abs, t1 rel 3D: thresholds for both medial 1-cells and 2-cells – t1 abs, t1 rel – t2 abs, t2 rel Thinning: removing simple pairs that are not medial cells – Note: only need to check the simple cell in a pair (the witness is never isolated)

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CSE554Cell ComplexesSlide 39 Choosing Thresholds Higher thresholds result in smaller skeleton – Threshold the absolute difference at ∞ will generally purge all cells at that dimension Except those for keeping the topology – Absolute threshold has more impact on features at small scales (e.g., noise) – Relative threshold has more impact on rounded features (e.g., blunt corners) t1 abs = ∞ Skeletons computed at threshold